The $m$-weak group orthogonality for operators

Year 2026, Volume: 55 Issue: 1 , 17 - 27 , 23.02.2026

Olivera Stanimirovic

https://doi.org/10.15672/hujms.1615369

https://izlik.org/JA94DG67MY

Abstract

The main goal is extending the concept of the core-EP orthogonality to the $m$-weak group orthogonality for bounded linear Drazin invertible Hilbert space operators, using the $m$-weak group inverse. Different properties and characterizations of $m$-weak group orthogonal operators are proved as well as their operator matrix forms. The connection between the $m$-weak group binary relation and the $m$-weak group orthogonality is given. We also study additive properties for the $m$-weak group inverse. Consequently, we study the weak group orthogonality for operators.

Keywords

m-weak group inverse , weak group inverse , core–EP orthogonality , core–EP pre-order

References

• [1] R. Behera, G. Maharana, J.K. Sahoo, Further results on weighted core-EP inverse of matrices, Results Math. 75, 174, 2020.
• [2] A. Ben-Israel, T.N.E. Grevile,Generalized inverses, theory and applications, Second edition, Canadian Mathematical Society, Springer, New York, Beflin, Heidelberg, Hong Kong, London, Milan, Paris, Tokyo, 2003.
• [3] G. Dolinar, B. Kuzma, J. Marovt, B. Ungor, Properties of core-EP order in rings with involution, Front. Math. China 14, 715–736, 2019.
• [4] D.E. Ferreyra, F.E. Levis, N. Thome, Revisiting the core EP inverse and its extension to rectangular matrices, Quaest. Math. 41(2), 265–281, 2018.
• [5] D. E. Ferreyra, S. Malik, Core and strongly core orthogonal matrices, Linear Multilinear Algebra 70(20), 5052–5067, 2022.
• [6] Y. Gao, J. Chen,Pseudo core inverses in rings with involution, Comm. Algebra 46(1), 38–50, 2018.
• [7] M. R. Hestenes, Relative Hermitian matrices, Pacific J. Math. 11, 224–245, 1961.
• [8] W. Jiang, K. Zuo, Further characterizations of the m-weak group inverse of a complex matrix, AIMS Mathematics 7(9), 17369–17392, 2022.
• [9] R. Kuang, C. Deng, Common properties among various generalized inverses and constrained binary relations, Linear Multilinear Algebra 71(8), 1295–1322, 2023.
• [10] W. Li, J. Chen, Y. Zhou, Characterizations and representations of weak core inverses and m-weak group inverses, Turk. J. Math. 47(5), 1453–1468, 2023.
• [11] X. Liu, K. Zhang, H. Jin, The m-WG inverse in the Minkowski space, Open Mathematics 21(1), 20230145, 2023.
• [12] I. Kyrchei, Determinantal representations of the core inverse and its generalizations with applications, Journal of Mathematics 2019, Article ID 1631979, 13 pages, 2023.
• [13] H. Ma, P.S. Stanimirovic, Characterizations, approximation and perturbations of the core-EP inverse, Appl. Math. Comput. 359, 404–417, 2019.
• [14] J. Marovt, D. Mosic, On some orders in $\ast$-rings based on the core-EP decomposition, J. Algebra Appl. 21(01), 2250010, 2022.
• [15] D. Mosic, Weighted core–EP inverse of an operator between Hilbert spaces, Linear Multilinear Algebra 67(2), 278–298, 2019.
• [16] D. Mosic, G. Dolinar, B. Kuzma, J. Marovt, Core–EP orthogonal operators, Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2022.2033155, 2022.
• [17] D. Mosic, P.S. Stanimirovic, L.A. Kazakovtsev, The m-weak group inverse for rectangular matrices, Electronic Research Archive (ERA) 32(3), 1822–1843, 2024.
• [18] D. Mosic, P.S. Stanimirovic, L.A. Kazakovtsev, Application of m-weak group inverse in solving optimization problems, Rev. R. Acad. Cienc. Exactas F´is. Nat. Ser. A Mat. RACSAM 118(1), 13, 2024.
• [19] D. Mosic, D. Zhang, New representations and properties of m-weak group inverse, Results Math. 78, 97, 2023.
• [20] D. Mosic, D. Zhang, Weighted weak group inverse for Hilbert space operators, Front. Math. China 15, 709–726, 2020.
• [21] K.M. Prasad, K.S. Mohana, Core-EP inverse, Linear Multilinear Algebra 62(6), 792–802, 2014.
• [22] H. Wang, J. Chen, Weak group inverse, Open Mathematics 16, 1218–1232, 2018.
• [23] H. Wang, X. Liu, The weak group matrix, Aequationes Math. 93, 1261–1273, 2019.
• [24] M. Zhou, J. Chen, Integral representations of two generalized core inverses, Appl. Math. Comput. 333, 187–193, 2018.
• [25] M. Zhou, J. Chen, Y. Zhou, Weak group inverses in proper $\ast$-rings , J. Algebra Appl. 19(12), 2050238, 2020.
• [26] Y. Zhou, J. Chen, M. Zhou, m-weak group inverses in a ring with involution, Rev. R. Acad. Cienc. Exactas F Nat. Ser. A Mat. RACSAM 115, 2, 2021.